for any \(M \in {{M}\mathrm {fd}}\) , where \(Q^{\prime } \colon {H}_\infty ^p \rightarrow {H}_\infty ^p\)  is the cofibrant replacement functor introduced in Appendix .
The functor \(\operatorname{hoRan}_\iota ^{\prime } \colon {H}_\infty ^{p \ell } \rightarrow \widetilde{{H}}_\infty ^{p \ell }\)  is a right Quillen functor, where on the target side we consider the Čech localisation with respect to open coverings—this is a consequence of [1]}.
It provides a concrete way of comparing sheaves on \({{C}\mathrm {art}}\)  to sheaves on \({{M}\mathrm {fd}}\) ; the functor
\(\operatorname{hoRan}_\iota ^{\prime } \colon {H}_\infty ^{p \ell } \rightarrow \widetilde{{H}}_\infty ^{p \ell }\) 
